The E8 root system

Currently # roots fixed, # moved by π/3, # by π/2, # by 2π/3, # by π.
Actions: reinitialize, scramble, Weyl word simplify (# moves).
Show current Weyl word, set a word, apply a word. Show cycle structure.
Apply sample Weyl group elements: of order 8 (centralizer of order 192), of order 8 (centralizer of order 64), of order 30 (centralizer of order 30), of order 24 (centralizer of order 24), of order 20 (centralizer of order 20), of order 18 (centralizer of order 108), of order 14 (centralizer of order 28).
Choose projection: default, alternate, squares, Petrie (8 30-gons), 10 24-gons, 12 20-gons, 13 (blurred) 18-gons, 7 (blurred) 14-gons; randomly perturb projection.
Coloring: recolor to projection, toggle color scheme, briefly show displacement angles.

Mathematical background

The display above shows a particular projection of the E8 root system, which can be described as a remarkable polytope in 8 dimensions (also known as the Gosset 421 polytope) having 240 vertices (known, in this context, as “roots”), and 6720 edges shown by black lines in the figure. Quite concretely, the roots can be described, in the coordinate system we have chosen, as the (112) points having coordinates (±1,±1,0,0,0,0,0,0) (where both signs can be chosen independently and the two non-zero coordinates can be anywhere) together with those (128) having coordinates (±½,±½,±½,±½,±½,±½,±½,±½) (where all signs can be chosen independently except that there must be an even number of minuses).

Every root (being identified with the vector leading from the origin to the vertex in question) is of the same length (i.e., all vertices are on a sphere with the origin as center; this is specific to E8 and is not a property of all root systems); the opposite of each root is again a root, and each one is orthogonal to 126 others, while forming an angle of π/3 with 56 others (those that are connected to it by an edge): the only possible angles between two roots are 0, π/3, π/2, 2π/3 and π.

The group of symmetries of this object is the group, known as the Weyl group of E8, generated by the (orthogonal) reflections about the hyperplane orthogonal to each root: this is a group of order 696729600 which can also be described as O8+(2). It is also the group of automorphism of the adjacency graph of the polytope.

Those 112 roots which have coordinates of the form (±1,±1,0,0,0,0,0,0) are shown as larger dots, and constitute a so-called D8 root system inside the E8 root system, which, as a polytope, is a rectified octacross; the reflections determined by those vertices generate a subgroup of order 5160960 (the Weyl group of D8, a subgroup of index 2 in {±1}≀𝕾8) of the full Weyl group of E8. The 128 remaining vertices (forming a demiocteract) are shown as smaller dots; alone, they are not a root system because the reflection determined by one of them does not fix that subset. Note that this division of the 240 vertices as 112+128 is particular to the chosen coordinate system and is not preserved by symmetries of the whole (except, precisely, by those living in the smaller Weyl group of D8; so there are 135 ways of making this decomposition).

One can further divide the roots in two by calling half of them “positive” in such a way that the sum of two positive roots, if it is a root, is always positive, and that for every root either it or its opposite is positive; there are many ways to do this (in fact, precisely as many as there are elements in the Weyl group), and we have chosen the division given by a lexicographic order on the coordinates: we call positive those roots such that the leftmost nonzero coordinate is positive (or, by numbering the roots lexicographically from 0 to 239, the positive ones are those numbered 120 through 239). A choice of positive roots is equivalent to a choice of fundamental (or simple) roots: these are the positive roots which cannot be written as a sum of two positive roots, and it then turns out that these form a basis of the ambient 8-space and, remarkably, that every positive root can be written as a linear combination of fundamental roots with nonnegative integer coefficients (equivalently, the fundamental roots form a non-orthogonal basis in which the coordinates of every root are either all nonnegative or all nonpositive; there is a uniquely defined greatest root, whose coordinates in terms of fundamental roots dominates that of every other root, and which happens to be one half the sum of all positive roots, fundamental or not: for E8, it is ⟨4,3,6,5,4,3,2,2⟩ and, for our choices, it is root number 239, or (1,1,0,0,0,0,0,0)). Any choice of positive/fundamental roots can be brought to any other choice by a unique element of the Weyl group.

If we represent the eight fundamental roots and connect two by a line whenever they form an angle of 2π/3 (the only other possibility being that they are orthogonal: in the case of E8, the angles of 3π/4 and 5π/6 do not occur), we obtain the so-called Dynkin diagram, which in the case of E8 has seven nodes in a simple chain and an eighth branching from the third. Here, we number the fundamental roots in the same total order as chosen to define the positive roots (i.e., lexicographic order on the coordinates; then the fundamental roots 1 through 8 are the roots numbered 120, 121, 122, 126, 132, 140, 150 and 162), and the Dynkin diagram has fundamental roots 8–1–3–4–5–6–7 in a chain and fundamental root number 2 branching off from 3.

The fundamental roots are important because the reflection with respect to them suffice to generate the Weyl group. Furthermore, the minimal length of an expression of a given element of the Weyl group as such a product of fundamental reflections (the length relative to the given element for the chosen system of fundamental roots) is equal to the number of positive roots whose image is a negative root; and composing by a fundamental reflection will always increase or decrease by 1 the length of the Weyl group element.